Solve for $a$, $ \dfrac{5a - 1}{a} = \dfrac{1}{3a} + \dfrac{9}{a} $
Solution: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $a$ $3a$ and $a$ The common denominator is $3a$ To get $3a$ in the denominator of the first term, multiply it by $\frac{3}{3}$ $ \dfrac{5a - 1}{a} \times \dfrac{3}{3} = \dfrac{15a - 3}{3a} $ The denominator of the second term is already $3a$ , so we don't need to change it. To get $3a$ in the denominator of the third term, multiply it by $\frac{3}{3}$ $ \dfrac{9}{a} \times \dfrac{3}{3} = \dfrac{27}{3a} $ This give us: $ \dfrac{15a - 3}{3a} = \dfrac{1}{3a} + \dfrac{27}{3a} $ If we multiply both sides of the equation by $3a$ , we get: $ 15a - 3 = 1 + 27$ $ 15a - 3 = 28$ $ 15a = 31 $ $ a = \dfrac{31}{15}$